JEE Main & Advanced Physics Simple Harmonic Motion Oscillation of Spring Combination

(1) Series combination : If two springs of spring constants \[{{K}_{1}}\] and \[{{K}_{2}}\] are joined in series as shown then

(i) In series combination equal forces acts on spring but extension in springs are different.

(ii) Spring constants of combination

\[\frac{1}{{{k}_{s}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}\]\[\Rightarrow \]\[{{k}_{s}}=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]

(iii) If n springs of different force constant are connected in series having force constant \[{{k}_{1}},{{k}_{2}},{{k}_{3}}.......\] respectively then 

\[\frac{1}{{{k}_{S}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}+\frac{1}{{{k}_{3}}}+........\]

If all spring have same spring constant then \[{{k}_{S}}=\frac{k}{n}\]

(iv) Time period of combination \[T=2\pi \sqrt{\frac{m}{{{k}_{S}}}}=2\pi \sqrt{\frac{m({{k}_{1}}+{{k}_{2}})}{{{k}_{1}}{{k}_{2}}}}\]

(2) Parallel combination : If the springs are connected in parallel as shown

(i) In parallel combination different forces acts on different  springs but extension in springs are same

(ii) Spring constants of combination \[{{k}_{P}}={{k}_{1}}+{{k}_{2}}\]      

(iii) If n springs of different force constant are connected in parallel having force constant \[{{k}_{1}},{{k}_{2}},{{k}_{3}}.......\] respectively then \[{{k}_{P}}={{k}_{1}}+{{k}_{2}}+{{k}_{3}}+...\]

If all spring have same spring constant then \[{{k}_{P}}=nk\]

(iv) Time period of combination \[{{T}_{P}}=2\pi \sqrt{\frac{m}{{{k}_{P}}}}=2\pi \sqrt{\frac{m}{({{k}_{1}}+{{k}_{2}})}}\]